Theorem itg2lecl

Description: If an S.2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014)

	Ref	Expression
Assertion	itg2lecl	⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		itg2cl	⊢ ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ* )
2	1	3ad2ant1	⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ* )
3		simp2	⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → 𝐴 ∈ ℝ )
4		itg2ge0	⊢ ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) → 0 ≤ ( ∫2 ‘ 𝐹 ) )
5	4	3ad2ant1	⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → 0 ≤ ( ∫2 ‘ 𝐹 ) )
6		simp3	⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → ( ∫2 ‘ 𝐹 ) ≤ 𝐴 )
7		xrrege0	⊢ ( ( ( ( ∫2 ‘ 𝐹 ) ∈ ℝ* ∧ 𝐴 ∈ ℝ ) ∧ ( 0 ≤ ( ∫2 ‘ 𝐹 ) ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ )
8	2 3 5 6 7	syl22anc	⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ∈ ℝ ∧ ( ∫2 ‘ 𝐹 ) ≤ 𝐴 ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ )